Siegel domain realization of pseudo-Hermitian symmetric manifolds
A description of some pseudo Hermitian symmetric spaces is given in the following way: Take \(Y\) a complex symmetric space (In general the complex Grassmann manifold \(\text{Gr}^ \mathbb{C}(2n,n)\) of complex linear \(n\) planes in \(\mathbb{C}^{2n}\), or a submanifold). The space \(X\) is an orbit of a noncompact real form acting on \(Y\). Some Zariski open set \(X_ 0\) of \(X\) (obtained by using inhomogeneous coordinates) is a tube domain over a nonconvex cone \(V\) which is homogeneous. Then, in the situation considered by the author, there exists some natural convex subcone \(V_ 1\) of \(V\) such that the associated tube domain \(X^ 1\) is a Hermitian symmetric space. Finally, the study of compact cycles is done. For instance, if \(Y=\text{Gr}^ \mathbb{C} (2n,n)\), one takes \(G=S1 (n,\mathbb{R})\) and the \(G\)-orbit associated is the orbit of \(Z_ 0=(I_ n, i I_ n)\). Then the isotropy group is isomorphic to \(Sl(n,\mathbb{C})\times t\) where \(T\) is the circle. The tube \(X_ 0\) associated to \(X\) has as basis the cone \(V\) of all real matrices of order \(n\) of positive determinant. The cone \(V_ 1\) is the cone of all positive definite matrices of order \(n\). A collection of compact complex submanifolds is also given.
scientific article; zbMATH DE number 205525
scientific article